What is the integral from 0 to 2 of x(1-1/2 x) ?

The integral from 0 to 2 of x(1-1/2 x) is 2/3

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integral from 0 to 2 of x(1-1/2 x)

\int_{0}^{2} x (1 - \frac{1}{2} x) d x

\frac{2}{3}

Decimal

0.66666 \dots

Solution steps

\int_{0}^{2} x (1 - \frac{1}{2} x) d x

Expand

x (1 - \frac{1}{2} x) : x - \frac{x ^{2}}{2}

= \int_{0}^{2} x - \frac{x ^{2}}{2} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{0}^{2} x d x - \int_{0}^{2} \frac{x ^{2}}{2} d x

\int_{0}^{2} x d x = 2

\int_{0}^{2} \frac{x ^{2}}{2} d x = \frac{4}{3}

= 2 - \frac{4}{3}

Simplify

2 - \frac{4}{3} : \frac{2}{3}

= \frac{2}{3}

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What is the integral from 0 to 2 of x(1-1/2 x) ?
The integral from 0 to 2 of x(1-1/2 x) is 2/3